On the \(p\)-part of the ideal class group of \(\mathbb{Q} (\zeta_ p + \zeta_ p^{-1})\) and Vandiver's conjecture
type: Item
instance of: scholarly article
title: On the \(p\)-part of the ideal class group of \(\mathbb{Q} (\zeta_ p + \zeta_ p^{-1})\) and Vandiver's conjecture
zbMATH Open document ID: 0844.11069
published in: Michigan Mathematical Journal
publication date: 1996-02-20T00:00:00Z
review text: Let \(p\geq 5\) be prime, and let \(\zeta\) denote a primitive \(p\)-th root of unity. Put \(K= \mathbb{Q} (\zeta)\) and \(G= \text{Gal} (K/Q)\); using the Teichmüller character \(\omega\) we can define idempotents \(e_0, \dots, e_{p-2}\) of \(\mathbb{Z}_p [G]\) corresponding to the powers of \(\omega\) and decompose the \(p\)-Sylow subgroup \(A\) of the ideal class group of \(K\) as well as the \(p\)-Sylow subgroup \(W\) of the unit group of \(K\) modulo the group of circular units into eigenspaces. Results of Mazur, Wiles and Kolyvagin state that, for even \(r\), the cardinalities of \(e_r (A)\) and of \(e_r (W)\) coincide and that, for \(r\) odd, \(\# e_r (A)\) is just the \(p\)-part of the generalized Bernoulli numbers. For fixed even \(r\geq 2\), define \(\beta= \beta_n= \prod_{k=1}^{p-1} (1- \zeta^k )^{k^{(p- 1-r) p^{n-1}}}\) and let \(\ell_n\) denote the largest integer \(\leq n\) such that \(\beta_n\) is a \(p^{\ell_n}\)th power in \(K\). It is known that the sequence \(\ell_n\) eventually stabilizes, and that the cardinalities of \(e_r (A)\) and of \(e_r (W)\) are equal to \(p^{\ell_n}\) for all large enough \(n\). It is these integers \(\ell_n\) which the author studies in this article. In fact, fix \(n\geq 1\) and let \(P_m\) denote the set of all prime ideals in \(K\) which are above the rational primes \(q\equiv 1\bmod p^n\) and which satisfy \(p^{ (q-1)/ p} \equiv \zeta \bmod Q\). Let \(s\) be a primitive root modulo \(Q\) such that \(s^{ (q-1) /p} \equiv \zeta \bmod Q\); then the index \(i(Q)\) defined by \(\beta \equiv s^{i (Q)} \bmod Q\) can be computed in terms of Gaussian periods, and we have \(p^{\ell_n} \leq p^{i (Q)}\) for all \(Q\in P_m\). Moreover, there exists a prime ideal \(Q\in P_m\) such that equality holds. This is the first of many results on the numbers \(i(Q)\) and \(\ell_n\) which are proved in this article but which are too technical to be given here. Suffice it to say that the author succeeds in proving a criterion which allows on…
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 838733
zbMATH Keywords: Vandiver's conjecture
zbMATH Keywords: real abelian field
zbMATH Keywords: annihilators of the \(p\)-class group
zbMATH Keywords: ideal class group
zbMATH Keywords: unit group
zbMATH Keywords: circular units
zbMATH Keywords: primitive root
zbMATH Keywords: index
zbMATH Keywords: Gaussian periods
author: Francisco Thaine
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029005231
OpenAlex ID: W1993224651
Wikidata QID: Q123200987
DOI: 10.1307/MMJ/1029005231
description: scientific article; zbMATH DE number 838733